If θ is positive acute angle and 3(sec2θ + tan2θ) = 5, then the value of cos2θ is?

Answer: Option A

$$\begin{array}{rl}& \text{3}\left({\sec }^2\theta +\text{ta}{\text{n}}^2\theta \right)=5\\ & {\sec }^2\theta +\text{ta}{\text{n}}^2\theta =\frac{5}{3}.....(i)\\ & {\sec }^2\theta -\text{ta}{\text{n}}^2\theta =1......(ii)\\ & \text{Adding equation (i) and (ii)}\\ & \text{2}{\sec }^2\theta =\frac{8}{3}\\ & sec\theta =\frac{2}{\sqrt{3}}\\ & \therefore \theta =30^{\circ }\\ & \cos 2\theta =\cos 2\left({30}^{\circ }\right)\\ & =\cos 60^{\circ }\\ & =\frac{1}{2}\end{array}$$